Main theorem of elimination theory: Difference between revisions

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Therefore, the ideal <math>\mathfrak r,</math> whose existence is asserted by the main theorem of elimination theory, is the zero ideal if {{math|''k'' < ''n''}}, and, otherwise, is generated by the maximal minors of the Macaulay matrix in degree {{mvar|D}}.

If {{math|1=''k'' = ''n''}}, Macaulay has also proved that, <math>\mathfrak r</math> is a [[principal ideal]] (although Macaulay matrix in degree {{mvar|D}} is not a square matrix when {{math|''k'' > 2}}), which is generated by the [[Macaulay's resultant|resultant]] of <math>\varphi(f_1),\ldots, \varphi(f_n).</math> This ideal is also [[generic property|generically]] a [[prime ideal]], as it is prime if {{mvar|R}} is the ring of [[integer polynomial]]s with the all coefficients of <math>\varphi(f_1),\ldots, \varphi(f_k)</math> as indeterminates.

==Geometrical interpretation==